The Rabbit Problem: Proof

Can we find a way to predict each number without going through each individual pair? Of course we can, we're mathematicians. The way you find the total number of rabbits for each month is to find out how many pairs of rabbits were newly born that month and add that to the number of rabbits you had before the new ones were born.

So how many pairs of rabbits are newly born every month? Well, since it takes two months for each new pair to give birth, each pair of rabbits that was alive two months ago will give birth to a new pair. In other words, the number of new pairs in each month is equal to the number of pairs alive two months ago.

Next we need to find the number of rabbit pairs that were alive before the new ones were born. It should be obvious that this is simply the number of pairs alive the month before.

In other words, to find the total number of pairs of rabbits, you simply add together the number of pairs that were alive in the preceding two months.

Now, do you know of any series of numbers which begins with one and one and continues by adding the preceding two numbers to get the next? Of course you do. It's the Fibonacci series.